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We introduce a wider class of bounded Hartogs domains, which contains somegeneralizations of the classical Hartogs triangle. A sharp criteria for the$L^p-L^q$ boundedness of the Toeplitz operator with symbol $K^{-t}$ is obtainedon these domains, where $K$ is the Bergman kernel on diagonal and $t\geq 0$. Itgeneralizes the results by Chen and Beberok in the case $1<p<\infty$.

Special Toeplitz operators on a class of bounded Hartogs domains